category theory

Contents

Definition

A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint $R$ (a cofree functor):

$C\stackrel{\stackrel{i}{↪}}{\underset{R}{←}}D\phantom{\rule{thinmathspace}{0ex}}.$C \stackrel{\overset{i}{\hookrightarrow}}{\underset{R}{\leftarrow}} D \,.

The dual concept is that of a reflective subcategory. See there for more details.

Properties

Theorem

Vopěnka's principle is equivalent to the statement:

For $C$ a locally presentable category, every full subcategory $D↪C$ which is closed under colimits is a coreflective subcategory.